In the Country of Madadia there are two major taxi companies, Green cabs and Blue cabs.

A cab was involved in a hit and run accident at night outside the "Sweaty Chef" kebab shop. Here is some data:

a) Although the two companies are equal in size, 85% of cab accidents in the city of Madville, one of Madadia's largest cities, involve Green cabs and 15% involve Blue cabs.

b) A witness identified the cab in this particular accident as Blue.

The Madville court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

If it looks like an obvious problem in statistics, then consider the following argument:

The probability that the colour of the cab was Blue is 80%! After all, the witness is correct 80% of the time, and this time he said it was Blue!
Is this right?

Answer

The police tests don't apply directly, because according to the wording, the witness, given any mix of cabs, would get the right answer 80% of the time.
Thus given a mix of 85% green and 15% blue cabs, he will say 20% of the green cabs and 80% of the blue cabs are blue.
That's 20% of 85% plus 80% of 15%, or 17%+12% = 29% of all the cabs that the witness will say are blue. Of those, only 12/29 are actually blue.
Thus P(cab is blue|witness claims blue) = 12/29.
That's just a little over 40%.Hide